Finite Group Databasebeta

← #10094

Subgroups — #10094

Order 256.

295 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3213C211yesmaximal, characteristic
2#3141C211yesmaximal, characteristic, complemented
3#2658C211yesmaximal, characteristic, complemented
4#1556C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic, complemented
6#1505C211yesmaximal, characteristic, complemented
7#1300C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8—21no—
10C2×C4⋊Q8C2211yescharacteristic
11C2×#124—21no—
12C2×#124C2211yescharacteristic
13#496—21no—
14#489—21no—
15#489—22no—
16#487—21no—
17#486—21no—
18#463—22no—
19#448—21no—
20#447—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#417—22no—
24C4×M16C2211yescharacteristic
25#399—22no—
26#399—22no—
27#376C2211yescharacteristic
28#332—22no—
29#329—22no—
30#327—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C22×Q8D411yescharacteristic
34C22×D4D411yescharacteristic
35C2×SD16—22no—
36C2×D8—22no—
37C2×M16—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—44no—
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#124—21no—
46#124—21no—
47#124—21no—
48#124—21no—
49#124—21no—
50#124—44no—
51#121—21no—
52#121—21no—
53C4×D4—21no—
54C4×D4—21no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D412yes—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63C4⋊C8—21no—
64C4⋊C8D412yes—
65#103—24no—
66#103—22no—
67#102—24no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71#97—21no—
72#97—21no—
73#97D412yes—
74C4×C8—21no—
75#95—21no—
76#95D412yes—
77C24—21no—
78#41—44no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8—42no—
83C2×Q8—42no—
84C2×Q8—42no—
85C2×Q8—41no—
86C2×Q8—41no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—42no—
93C2×D4—41no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—41no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101SD16—82no—
102D8—82no—
103M16—42no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—41no—
107C4⋊C4—22no—
108C4⋊C4—21no—
109C4⋊C4—22no—
110C4⋊C4—21no—
111C4⋊C4—48no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C22⋊C4—48no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C42—24no—
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C42#4111yescharacteristic
126C23—42no—
127C23—42no—
128C23—41no—
129C23—41no—
130C23—42no—
131C23#12011yescharacteristic
132Q8—42no—
133Q8—42no—
134Q8—81no—
135Q8—82no—
136Q8—82no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141D4—81no—
142D4—82no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—82no—
156C2×C4#12011yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12711yescharacteristic
162C8—42no—
163C8—42no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—84no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22#39211yescharacteristic
175C22#45412yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—44no—
182C4—81no—
183C4—82no—
184C4—81no—
185C4#45312yes—
186C2—22no—
187C2—161no—
188C2—161no—
189C2#200211yescharacteristic
190C2#200311yescharacteristic
191C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3213C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3141C2semidirect
3. #2658C2semidirect
4. #1556C2semidirect
5. #1527C2semidirect
6. #1505C2semidirect
7. #1300C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#32132
#31412
#26582
#15562
#15272
#15052
#13002